edu.jas.integrate
Class ElementaryIntegration<C extends GcdRingElem<C>>

java.lang.Object
  edu.jas.integrate.ElementaryIntegration<C>

Type Parameters:

C - coefficient type

public class ElementaryIntegration<C extends GcdRingElem<C>>
extends java.lang.Object

Methods related to elementary integration. In particular there are methods for Hermite reduction and Rothstein-Trager integration of the logarithmic part.

Author:

Axel Kramer, Heinz Kredel

Field Summary

FactorAbstract<C>	irr Engine for factorization.
SquarefreeAbstract<C>	sqf Engine for squarefree decomposition.
GreatestCommonDivisorAbstract<C>	ufd Engine for greatest common divisors.

Constructor Summary

ElementaryIntegration(RingFactory<C> br)
Constructor.

Method Summary

Quotient<C>	deriviative(Quotient<C> r) Derivation of a univariate rational function.
Integral<C>	integrate(GenPolynomial<C> a, GenPolynomial<C> d) Integration of a rational function.
QuotIntegral<C>	integrate(Quotient<C> r) Integration of a rational function.
java.util.List<GenPolynomial<C>>[]	integrateHermite(GenPolynomial<C> a, GenPolynomial<C> d) Integration of the rational part, Hermite reduction step.
LogIntegral<C>	integrateLogPart(GenPolynomial<C> A, GenPolynomial<C> P) Univariate GenPolynomial integration of the logaritmic part, Rothstein-Trager algorithm.
LogIntegral<C>	integrateLogPartIrreducible(GenPolynomial<C> A, GenPolynomial<C> P) Univariate GenPolynomial integration of the logaritmic part, Rothstein-Trager algorithm.
boolean	isIntegral(LogIntegral<C> rl) Test of integration of the logarithmic part of a rational function.
boolean	isIntegral(QuotIntegral<C> ri) Test of integration of a rational function.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

irr

public final FactorAbstract<C extends GcdRingElem<C>> irr

Engine for factorization.

sqf

public final SquarefreeAbstract<C extends GcdRingElem<C>> sqf

Engine for squarefree decomposition.

ufd

public final GreatestCommonDivisorAbstract<C extends GcdRingElem<C>> ufd

Engine for greatest common divisors.

Constructor Detail

ElementaryIntegration

public ElementaryIntegration(RingFactory<C> br)

Constructor.

Method Detail

integrate

public QuotIntegral<C> integrate(Quotient<C> r)

Integration of a rational function.

Parameters:

r - rational function

Returns:

Integral container, such that integrate(r) = sum_i(g_i) + sum_j( an_j log(hd_j) )

integrate

public Integral<C> integrate(GenPolynomial<C> a,
                             GenPolynomial<C> d)

Integration of a rational function.

Parameters:

a - numerator

d - denominator

Returns:

Integral container, such that integrate(a/d) = sum_i(gn_i/gd_i) + integrate(h0) + sum_j( an_j log(hd_j) )

integrateHermite

public java.util.List<GenPolynomial<C>>[] integrateHermite(GenPolynomial<C> a,
                                                           GenPolynomial<C> d)

Integration of the rational part, Hermite reduction step.

Parameters:

a - numerator

d - denominator, gcd(a,d) == 1

Returns:

[ [ gn_i, gd_i ], [ h0, hn_j, hd_j ] ] such that integrate(a/d) = sum_i(gn_i/gd_i) + integrate(h0) + sum_j( integrate(hn_j/hd_j) )

integrateLogPart

public LogIntegral<C> integrateLogPart(GenPolynomial<C> A,
                                       GenPolynomial<C> P)

Univariate GenPolynomial integration of the logaritmic part, Rothstein-Trager algorithm.

Parameters:

A - univariate GenPolynomial, deg(A) < deg(P).

P - univariate squarefree GenPolynomial, gcd(A,P) == 1.

Returns:

logarithmic part container.

integrateLogPartIrreducible

public LogIntegral<C> integrateLogPartIrreducible(GenPolynomial<C> A,
                                                  GenPolynomial<C> P)

Univariate GenPolynomial integration of the logaritmic part, Rothstein-Trager algorithm.

Parameters:

A - univariate GenPolynomial, deg(A) < deg(P).

P - univariate irreducible GenPolynomial. // gcd(A,P) == 1 automatic

Returns:

logarithmic part container.

deriviative

public Quotient<C> deriviative(Quotient<C> r)

Derivation of a univariate rational function.

Parameters:

r - rational function

Returns:

dr/dx

isIntegral

public boolean isIntegral(QuotIntegral<C> ri)

Test of integration of a rational function.

Parameters:

ri - integral

Returns:

true, if ri is an integral, else false.

isIntegral

public boolean isIntegral(LogIntegral<C> rl)

Test of integration of the logarithmic part of a rational function.

Parameters:

rl - logarithmic part of an integral

Returns:

true, if rl is an integral, else false.